Abstract
By a degree of a module M we mean a numerical measure of information carried by M. It must serve the purposes of allowing comparisons between modules and to exhibit flexible calculus rules that track the degree under some basic constructions in module theory.
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References
G. Almkvist, K-theory of endomorphisms, J. Algebra 55 (1978), 308–340; Erratum, J. Algebra 68 (1981), 520–521.
D. Bayer and D. Mumford, What can be computed in Algebraic Geometry?, in Computational Algebraic Geometry and Commutative Algebra, Proceedings, Cortona 1991 ( D. Eisenbud and L. Robbiano, Eds. ), Cambridge University Press, 1993, 1–48.
D. Bayer and M. Stillman, Macaulay: A system for computation in algebraic geometry and commutative algebra, 1992. Available via anonymous ftp from math.harvard.edu.
J. Becker, On the boundedness and unboundedness of the number of generators of ideals and multiplicity, J. Algebra 48 (1977), 447–453.
M. Boratynski, D. Eisenbud and D. Rees, On the number of generators of ideals in local Cohen-Macaulay rings, J. Algebra 57 (1979), 77–81.
W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge University Press, Cambridge, 1993.
A. Capani, G. Niesi and L. Robbiano, CoCoA: A system for doing computations in commutative algebra, 1995. Available via anonymous ftp from lancelot.dima.unige.it.
L. R. Doering, Multiplicities, Cohomological Degrees and Generalized Hilbert Functions, Ph.D. Thesis, 1997, Rutgers University.
L. R. Doering, T. Gunston and W. V. Vasconcelos, Cohomological degrees and Hilbert functions of graded modules, Preprint, 1996.
P. Eakin and A. Sathaye, Prestable ideals, J. Algebra 41 (1976), 439–454.
D. Eisenbud, Commutative Algebra with a view toward Algebraic Geometry, Springer-Verlag, Berlin-Heidelberg-New York, 1995.
D. Eisenbud and S. Goto, Linear free resolutions and minimal multiplicities, J. Algebra 88 (1984), 89–133.
A. Grothendieck, Local Cohomology, Lecture Notes in Mathematics 41, Springer-Verlag, Berlin-Heidelberg-New York, 1967.
T. H. Gulliksen, On the length of faithful modules over Artinian local rings, Math. Scand. 31 (1972), 78–82.
T. Gunston, Ph.D. Thesis in Progress, Rutgers University.
R. Hartshorne, Connectedness of the Hilbert scheme, Publications Math. I.H.E.S. 29 (1966), 261–304.
J. Herzog and E. Kunz, Der kanonische Modul eines Cohen-Macaulay Rings, Lecture Notes in Mathematics 238, Springer-Verlag, Berlin-Heidelberg-New York, 1971.
L. T. Hoa, Reduction numbers of equimultiple ideals, J. Pure ê9 Applied Algebra 109 (1996), 111–126.
C. Huneke and B. Ulrich, General hyperplane sections of algebraic varieties, J. Algebraic Geometry 2 (1993), 487–505.
D. Kirby and D. Rees, Multiplicities in graded rings I: The general theory, Contemporary Math. 159 (1994), 209–267.
C. Lech, Note on multiplicities of ideals, Arkiv far Matematik 4, (1960), 63–86.
F. S. Macaulay, The Algebraic Theory of Modular Systems, Cambridge University Press, Cambridge, 1916.
F. S. Macaulay, Some properties of enumeration in the theory of modular systems, Proc. London Math. Soc. 26 (1927), 531–555.
H. Matsumura, Commutative Algebra, Benjamin/Cummings, Reading, 1980.
C. Miyazaki, K. Yanagawa and W. Vogel, Associated primes and arithmetic degree, J. Algebra, to appear.
M. Nagata, Local Rings, Interscience, New York, 1962.
J. D. Sally, Bounds for numbers of generators for Cohen-Macaulay ideals, Pacific J. Math. 63 (1976), 517–520.
J. D. Sally, Number of Generators of Ideals in Local Rings, Lecture Notes in Pure & Applied Math. 35, Marcel Dekker, New York, 1978.
I. Schur, Zur Theorie der vertauschbaren Matrizen, J. reine angew. Math. 130 (1905), 66–76.
A. Shalev, On the number of generators for ideals in local rings, Advances in Math. 59 (1986), 82–94.
R. P. Stanley, Hilbert functions of graded algebras, Advances in Math. 28 (1978), 57–83.
R. P. Stanley, On the Hilbert function of a graded Cohen-Macaulay domain, J. Pure e.4 Applied Algebra 73 (1991), 307–314.
J. StĂĽckrad and W. Vogel, Buchsbaum Rings and Applications, Springer-Verlag, Vienna-New York, 1986.
B. Sturmfels, N. V. Trung and W. Vogel, Bounds on degrees of projective schemes, Math. Annalen 302 (1995), 417–432.
N. V. Trung, Reduction exponent and degree bound for the defining equations of graded rings, Proc. Amer. Math. Soc. 101 (1987), 229–236.
N. V. Trung, Bounds for the minimum number of generators of generalized Cohen—Macaulay ideals, J. Algebra 90 (1984), 1–9.
G. Valla, Generators of ideals and multiplicities, Comm. in Algebra 9 (1981), 1541–1549.
W. V. Vasconcelos, Arithmetic of Blowup Algebras, London Math. Soc., Lecture Note Series 195, Cambridge University Press, Cambridge, 1994.
W. V. Vasconcelos, The reduction number of an algebra, Compositio Math. 104 (1996), 189–197.
W. V. Vasconcelos, The homological degree of a module, Trans. Amer. Math. Soc.,to appear.
W. V. Vasconcelos, Computational Methods in Commutative Algebra and Algebraic Geometry, Springer-Verlag, to appear in 1997.
C. Weibel, An Introduction to Homological Algebra, Cambridge University Press, Cambridge, 1994.
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Vasconcelos, W.V. (1998). Cohomological Degrees of Graded Modules. In: Elias, J., Giral, J.M., Miró-Roig, R.M., Zarzuela, S. (eds) Six Lectures on Commutative Algebra. Progress in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0346-0329-4_6
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